Methods for improving the optical properties of bragg gratings

ABSTRACT

Methods to improve the optical properties of Bragg gratings are disclosed. A first method includes a post correction of the refractive index profile by applying an average index correction thereto. The average index correction is obtained through an analysis of the defects of the refractive index profile characterised through a reconstruction thereof. A second method includes a pre-correction to the refractive index profile by characterising the defects of a test grating, and again calculating an average index correction based thereon. Further gratings are then made using a corrected refractive index profile.

FIELD OF THE INVENTION

[0001] The present invention relates to the making of Bragg gratings andmore particularly concerns methods to compensate for defects found incurrently manufactured Bragg gratings.

BACKGROUND OF THE INVENTION

[0002] Bragg grating technology has been a subject of intensive researchfor the last decade, especially for its application in thetelecommunication industry. Although the optical performance of Bragggratings is high and extremely attractive for many applications, someproblems still persist such as the well known ripples in the group delayspectrum, especially observed in gratings for dispersion compensation.

[0003] Various approaches have been used to further improve the opticalproperties of Bragg gratings, and in particular minimize the group delayripples (GDR). Bragg gratings are usually made by photoinducing in anoptical fiber, or other photosensitive medium, a diffraction pattern,produced by actinic radiation projected through a phase mask. Animportant source of defects in the resulting grating is the mask itselfwhich usually contains phase errors due to its own manufacturingprocess. Efforts have therefore been made to improve the quality ofphase masks, and therefore reduce the defects in the resulting Bragggratings. For example, KOMUKAI et al (“Group delay ripple reduction andreflectivity increase in a chirped fiber Bragg grating bymultiple-overwriting of a phase mask with an electron-beam,” IEEEPhoton. Technol. Lett., vol. 12, pp. 816-818 (2000)) discloses astep-chirped phase mask made by overwriting a pattern at the same placeon a substrate several times, using an electron-beam in a continuousmovement approach. The same authors have also suggested another strategyfor fabricating a step-chirped phase mask using a raster scan-typelaser-beam writing system (T. Komukai, T. Inui, M. Kurihara, and S.Fujimoto, “Group-Delay Ripple Reduction in Step-Chirped Fiber BraggGratings by Using Laser-Beam Written Step-Chirped Phase Masks,” IEEEPhoton. Technol. Lett., vol. 14, pp. 1554-1556 (2002)). It is notablethat, as they modify the masks themselves, both these methods will onlycorrect for mask-related, systematic defects in the resulting gratings.

[0004] Instead of trying to minimize or eliminate phase errors in phasemasks, another approach is to minimize their effect on the resultinggrating, by a proper characterization of the origin of systematicdefects in the gratings produced by a given system, and an appropriatefeed-back on the fabrication process. Such a technique is for exampleshown in published U.S. patent application no. 2003/0059164 (STEPHANOVet al.) and A. V. Buryak, and D. Y. Stepanov, “Correction of systematicerrors in the fabrication of fiber Bragg gratings,” Opt. Lett., vol. 27,pp. 1099-1101 (2002). STEPHANOV et al. discloses a method forcompensating for phase errors in a Bragg grating by first making a testgrating, and then measuring its spectral characteristics. Thesecharacteristics are used to reconstruct the actual design of thegrating, preferably using a Layer Peeling Method. A compensated designis obtained by comparing the reconstructed design to the theoreticalstructure, for example by direct subtraction of deviations therebetween,and the compensated design is finally used to make subsequent gratingsusing the same optical system. Of course, only systematic defectsinherent to the particular optical system used to make the gratings willbe compensated for by this method.

[0005] Another alternative is to apply a post-treatment to thephotoinduced grating. Post-correction of Bragg gratings was alreadyproposed for other purposes, such as tuning the dispersion or otheroptical characteristics of the grating (see for example K. O. Hill etal. “Chirped in-fibre Bragg grating dispersion compensators:Linearisation of dispersion characteristic and demonstration ofdispersion compensation in 100 km, 10 Gbit/s optical fibre link,”Electron. Lett., vol. 30, pp. 1755-1756 (1994); and K. O. Hill et al.“Chirped in-fiber Bragg gratings for compensation of optical-fiberdispersion,” Opt. Lett., vol. 19, pp. 1314-1316 (1994)). When applied tothe correction of defects, this approach has the advantage ofalleviating both systematic and non-systematic errors. Referring topublished U.S. patent application no. 2003/0186142 (DESHMUKH et al.) andM. Sumetsky et al, “Reduction of chirped fiber grating group delayripple penalty through UV post processing,” Tech. Dig. Post deadlinepapers, OFC'2003, PD28, there is shown such a technique. The Bragggrating is photoinduced in a photosensitive medium, and a test beam islaunched in this medium during, or at the end of the writing process, tooptically characterise the grating. The collected data is used tocalculate a post-correction to the grating, using a correction algorithmbased on a simple solution to the inverse problem relating the measuredGDR vs. wavelength to the desired change in Bragg wavelength vs.position. In this approach, only the low frequency part of the GDR iscompensated for, which means that the technique only corrects for largedefects in the Bragg grating, that is about 10 mm or higher. Severaliterations can be made to optimize the benefit of this technique; afterthe correction is applied, the optical properties of the grating areagain measured, and a new correction calculated, this process beingrepeated until a satisfying suppression of the GDR ripples is achieved.

[0006] In spite of all the above-mentioned work, there is still a needfor defects-correction techniques having improved optical performances.In particular, there is a need for a technique that takes intoconsideration any or all of the types of defects found in Bragg gratingsand appropriately compensates therefore.

OBJECTS AND SUMMARY OF THE INVENTION

[0007] A first object of the present invention is to provide a methodfor improving the optical characteristics of Bragg gratings by adding apost-correction step in the fabrication procedure.

[0008] Another object of the invention is to apply such apost-correction which can successfully correct both systematic andnon-systematic errors.

[0009] In accordance with the first object of the invention, there isprovided a method for improving optical properties of a Bragg gratinghaving a spatial refractive index profile along a propagation axis. Themethod includes the following steps:

[0010] a) Characterising defects of the spatial refractive index profileof the Bragg grating. This characterising includes the sub-steps of:

[0011] i. Measuring optical properties of the grating;

[0012] ii. Reconstructing the spatial refractive index profile of thegrating based on these measured optical properties; and

[0013] iii. Comparing the reconstructed spatial refractive index profilewith a target spatial refractive index profile;

[0014] b) Calculating an average index correction to the spatialrefractive index profile as a function of the defects characterised instep a); and

[0015] c) Applying this average index correction to the Bragg grating.

[0016] Preferably, the defects characterised in step a) are perioddefects, apodization defects or both. Although average index defects canalso exist, they are virtually undistinguishable from period (or phase)defects and can be ignored.

[0017] A second object of the invention is to provide a pre-correctionstep in the fabrication procedure of the Bragg grating, the correctionserving as a feedback to the fabrication process for a series of othergratings in cases where only systematic errors are to be compensatedfor.

[0018] In accordance with this second aspect of the invention, there istherefore also provided a method for making an improved Bragg gratingusing an optical system generating systematic defects, which includesthe steps of:

[0019] a) making at least one test Bragg grating using theabove-mentioned optical system, which is set up to produce a targetspatial refractive index profile;

[0020] b) Characterising period defects and apodization defects of saidtest Bragg grating to respectively obtain a period defects functionδp(z) and an apodization defects function δ_(n)(z);

[0021] c) Calculating an average index correction to the target spatialrefractive index profile as a function of the period and apodizationdefects functions;

[0022] d) Calculating a corrected spatial refractive index profile usingthe average index correction; and

[0023] e) Making the improved Bragg grating using the optical system,set up to produce the corrected spatial refractive index profile.

[0024] In either case, the correction can advantageously be applied tosophisticated gratings such as multi-channel gratings or low-dispersionWDM Bragg gratings, as well as simple single-channel apodized gratings.For multi-channel gratings, only the defects affecting all the channelsmay be corrected for.

[0025] Further features and advantages of the present invention will bebetter understood upon reading of preferred embodiments thereof withreference to the appended drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0026]FIGS. 1A and 1B (PRIOR ART) are graphical representations of theapodisation and period profiles, respectively, obtained from a LayerPeeling reconstruction in comparison with the theoretical profiles(dotted lines).

[0027]FIG. 2 (PRIOR ART) is a graphical representation of the centralreflection location as a function of the wavelength for the grating andfor single-frequency parasitic reflections.

[0028]FIGS. 3A (PRIOR ART) and 3B are graphical representations of GroupDelay Ripples as a function of wavelength for a dispersion compensationgrating having a dispersion of −850 ps/nm, respectively before and afterthe application of a post-correction according to a preferred embodimentof the present invention.

[0029]FIGS. 4A to 4D (PRIOR ART) respectively show the reflectionspectrum (dB), transmission spectrum (dB), group delay (ps) and GDR as afunction of wavelength of a 4-channel dispersion compensation gratingmade using a non-corrected target period profile.

[0030]FIGS. 5A to 5D (PRIOR ART) respectively show the reflectionspectrum (dB), transmission spectrum (dB), group delay (ps) and GDR as afunction of wavelength of a 4-channel dispersion compensation gratingmade using a corrected target period profile according to anotherpreferred embodiment of the invention.

[0031]FIG. 6 is a schematic representation of an optical system forapplying an average index correction to a Bragg grating according to apreferred embodiment of the invention.

[0032]FIG. 7 (PRIOR ART) is a side view of an optical system to writetest gratings and improved gratings according to a preferred embodimentof the invention.

DESCRIPTION OF PREFERRED EMBODIMENTS OF THE INVENTION

[0033] The present invention provides methods for improving the opticalproperties of Bragg gratings in general, by calculating and applying anaverage index correction which compensates for defects in the gratings.To better understand the principles of the invention, it is relevant todefine and mathematically describe what is understood by “defects” inthe grating.

[0034] By definition, a Bragg grating consists of a modulation of theindex of refraction along a propagation axis of a waveguide. Intelecommunications applications the waveguide is usually a length ofoptical fiber, but it is understood that the present invention may beembodied in a planar waveguide or any other forms of optical medium inwhich a light beam may be guided. Throughout the description below, thepropagation axis is denoted as the z axis but it will be readilyunderstood that it is simply a convenient label.

[0035] A Bragg grating can be modelled generally as:

n ₀(z)=n _(eff0)(z)+Δn ₀(z)·sin(φ₀(z)),  (1)

[0036] where n_(eff0)(z) is the average effective index of the medium,Δn₀(z) is the index modulation (or “apodization” profile) and φ₀(z)represents the absolute phase profile of the grating. The profilen_(eff0)(z) is qualified of “average index” even if it is a spatiallydependent function, as it contains no short-period modulation; it isconsidered to vary slowly along z, the term “average” being understoodto be on a spatial scale of the grating fringes.

[0037] Alternatively, Equation (1) can be written as: $\begin{matrix}{{{n_{0}(z)} = {{n_{{eff}\quad 0}(z)} + {\Delta \quad {{n_{0}(z)}\quad \cdot \quad \sin}\quad ( {{\frac{2\pi}{p_{av}} \cdot \quad z} + {{\Delta\varphi}_{0}(z)}} )}}},} & (2)\end{matrix}$

[0038] where p_(av) represents the average period and Δφ₀(z) is thephase of the grating which describes the variations of the period withrespect to its average value p_(av). It is assumed that Δφ₀(z) is aslowly varying function, or in other words, that φ₀(z) in equation (1)is approximately a linear function of z. Although the representation ofa grating by Equation (2) is widely found in the literature, therepresentation by Equation (1) will be preferred in this document, asthe use of the absolute phase φ₀(z) is more convenient to describe theprinciples of the present invention.

[0039] The local period p₀(z) of the grating is defined as the distancealong the propagation axis of the grating over which a phase difference2π exists. In other words, the local period p₀(z) is defined as:$\begin{matrix}{{p_{0}(z)} \equiv {2\pi \quad {( \frac{d\quad {\varphi_{0}(z)}}{dz} )^{- 1}.}}} & (3)\end{matrix}$

[0040] It will be noted that in Equations (1) to (3), as well asthroughout the present description, the indices 0 are not related tosome point of origin, but refer to perfect or ideal quantities, that is,what these quantities would be if the grating had no defects.

[0041] An imperfect grating that contains defects can be modelled as:

n(z)=n _(eff)(z)+Δn(z)·sin(φ(z)),  (4)

[0042] where

n _(eff)(z)=n _(eff0)(z)+δn_(eff)(z),  (5)

Δn(z)=Δn ₀(z)·(1+δ_(n)(z)),  (6)

φ(z)=φ₀(z)+δφ(z),  (7)

[0043] and where δn_(eff)(z), δ_(n)(z) and δφ(z) are functionsrepresenting the local average index defects, apodisation (or modulationindex) defects and phase defects, respectively. To improve the opticalproperties of an imperfect grating, one must therefore minimize ornegate the effect of these three terms on the refractive index profileof the grating.

[0044] To change the optical properties of a Bragg grating, one mayeither effect a post-correction, that is an additional processing of analready written grating, or a pre-correction, which involves using themeasured properties of a test grating to modify the writing set up toimprove subsequent gratings obtained therewith. In the case of apost-correction, the grating phase profile can not be modified, that is,the physical locations where clear and dark fringes are along thegrating axis can not be changed. It is therefore not possible to changeφ(z). The apodisation profile, that is Δn(z) could be modified bysuperimposing a correction grating to the already written grating butthis approach is difficult to successfully achieve. Only the directmodification of the average index n_(eff)(z) is easy to realize. It istherefore advantageous to calculate an average index correction takinginto account the three types of defects of the grating, giving the finalgrating profile:

n(z)=n _(eff)(z)+Δn(z)·sin(φ(z))+Δn _(eff) _(—) _(tc)(z).  (8)

[0045] The correction to be calculated is therefore an average indexfunction Δn_(eff) _(—) _(tc)(z). As expressed by Equation (4), threekinds of defects have to be considered to completely model a Bragggrating: the average index, phase, and apodisation defects. It will beshown below that an average index correction function can be calculatedfor each of them. It is assumed that an appropriate total correctionfunction Δn_(eff) _(—) _(tc)(z) is obtained by summing the correctionscalculated for each kind of defect:

Δn _(eff) _(—) _(tc)(z)=Δn _(eff) _(—) _(cn)(z)+Δn _(eff) _(—)_(cp)(z)+Δn _(eff) _(—) _(ca)(z)+Δn _(eff) _(—) _(offset),  (9)

[0046] where Δn_(eff) _(—) _(cn)(z), Δn_(eff) _(—) _(cp)(z) and Δn_(eff)_(—) _(ca)(z) are the correction functions associated with the averageindex, phase and apodisation defects respectively, and where Δn_(eff)_(—) _(offset) is a uniform index change offset that renders thecorrection function Δn_(eff) _(—) _(tc)(z) strictly positive for allpositions, while Δn_(eff) _(—) _(cn)(z), Δn_(eff) _(—) _(cp)(z) andΔn_(eff) _(—) _(ca)(z) could be negative at some points. It will beunderstood that for post-correction of the grating the total correctionfunction has to be positive in order to be physically applicable, due tothe nature of the photosensitivity that causes an increase of therefractive index. In the case of a pre-correction, the total correctionfunction could be negative and the offset term could be omitted.

[0047] Correction of the Average Index Defects

[0048] A grating that contains only average index defects can bemodelled as:

n(z)=n_(eff0)(z)+δn_(eff)(z)+Δn₀(Z)·sin(φ₀(z)).  (10)

[0049] The average index correction function Δn_(eff) _(—) _(cn)(z) issimply the one that nulls the local average index defects δn_(eff)(z):

Δn _(eff) _(—) _(cn)(z)=−δn_(eff)(z).  (11)

[0050] Correction of the Phase Defects

[0051] A grating that contains only phase defects can be modelled as:

n(z)=n_(eff0)(z)+Δn₀(z)·sin(φ₀(z)+δφ(z)),  (12)

[0052] where δφ(z) is the phase defects function. The relationshipbetween the local period p(z) and the absolute phase, as expressed byEquation (3), becomes in this particular case: $\begin{matrix}{{p(z)} = {2\pi \quad {( {\frac{d}{dz}( {{\varphi_{0}(z)} + {{\delta\varphi}(z)}} )} )^{- 1}.}}} & (13)\end{matrix}$

[0053] Due to the phase defects δφ(z), the period profile is differentfrom the ideal profile p₀(z) by a certain quantity δp(z):

p(z)=p ₀(z)+δp(z).  (14)

[0054] Assuming that the spatial variation of δφ(z) is small compared tothe spatial variation of φ₀(z), the combination of Equations (3), (13)and (14) yields a period defects function δp(z) given by:$\begin{matrix}{{\delta \quad {p(z)}} = {{{- \frac{{p_{0}}^{2}(z)}{2\pi}} \cdot \frac{d}{dz}}{( {{\delta\varphi}(z)} ).}}} & (15)\end{matrix}$

[0055] Thus, there is equivalence between phase errors and perioderrors, which is explicitly expressed by Equation (15).

[0056] There is also equivalence between period and average indexerrors. Indeed, the local optical period length, that is, the product ofthe local values of period and average index, is the parameter thataffects the optical properties. Two different gratings for which, ateach point, the product of the period by the average index is the same,will virtually have the same optical characteristics. As an example,from the spectral characteristics, a Layer Peeling algorithm can notdistinguish between period and average index fluctuations. Using such areconstruction algorithm, all the fluctuations are attributed to theperiod and the average index is assumed to be uniform along the grating.

[0057] Accordingly, a grating that contains period errors δp(z) behavesas a grating that contains average index errors Δn_(eff) _(—) _(p)(z)if, at each point, their optical period lengths are the same, that is:

p ₀(z)·(n _(eff0)(z)+Δn _(eff) _(—) _(p)(z))=(p ₀(z)+δp(z))·n_(eff0)(z).  (16)

[0058] In other words, index errors Δn_(eff) _(—) _(p)(z) and perioderrors δp(z) are equivalent if they satisfy: $\begin{matrix}{\frac{\Delta \quad {n_{{eff}\quad \_ \quad p}(z)}}{n_{{eff}\quad 0}(z)} = {\frac{\delta \quad {p(z)}}{p_{0}(z)}.}} & (17)\end{matrix}$

[0059] Therefore, an average index correction function Δn_(eff) _(—)_(cp)(z) nulls the period errors δp(z)(or phase errors δφ(z)) if itsatisfies: $\begin{matrix}{{\Delta \quad {n_{{eff}\quad \_ \quad {cp}}(z)}} = {{{- \Delta}\quad {n_{{eff}\quad \_ \quad p}(z)}} = {{{{- \frac{n_{{eff}\quad 0}(z)}{p_{0}(z)}}\quad \cdot \quad \delta}\quad {p(z)}} = {{{n_{{eff}\quad 0}(z)}\quad \cdot \quad \frac{p_{0}(z)}{2\pi} \cdot \frac{d}{dz}}{( {\delta \quad {\varphi (z)}} ).}}}}} & (18)\end{matrix}$

[0060] A period correction function Δn_(eff) _(—) _(cp)(z), that is, anaverage index correction as a function of the period or phase defectscan therefore be calculated using equation (18) above.

[0061] Correction of the Apodization Defects

[0062] The correction of apodisation defects using an average indexcorrection function is less straightforward than the other two kinds ofdefects and preferably involves an analysis of the apodization defectfunction in the frequency domain.

[0063] A grating that contains only apodisation defects can be modelledas:

n(z)=n_(eff0)(z)+Δn₀(z)·(1+δ_(n)(z))·sin(φ₀(z)),  (19)

[0064] where δ_(n)(z) is the apodization defects function. Using afrequency domain analysis, δ_(n)(z) can be written as the sum ofperiodic error components: $\begin{matrix}{{{\delta_{n}(z)} = {\int_{0}^{\infty}{( {{{A_{c}(\alpha)}\quad \cos \quad ({\alpha z})} + {{A_{s}(\alpha)}{\sin ({\alpha z})}}} )\quad {\alpha}}}},} & (20)\end{matrix}$

[0065] where A_(c)(α) and A_(s)(α) are Fourier coefficients.

[0066] Combining Equations (19) and (20) yields to:

n(z)=n_(eff0)(z)+Δn ₀(z)·sin(φ₀(z))+Δn_(b)(z)+Δn_(r)(z),  (21)

[0067] where $\begin{matrix}{{{\Delta \quad {n_{b}(z)}} = {\int_{0}^{\infty}{( {{{\frac{\Delta \quad {{n_{0}(z)} \cdot {A_{c}(\alpha)}}}{2} \cdot \sin}\quad ( {{\varphi_{0}(z)} + {\alpha \quad z}} )} - {\frac{\Delta \quad {{n_{0}(z)} \cdot {A_{s}(\alpha)}}}{2}\quad \cdot \quad {\cos( {{\varphi_{0}(z)} + {az}}\quad )}}} ){\alpha}}}},} & (22) \\{{\Delta \quad {n_{r}(z)}} = {\int_{0}^{\infty}{( {{{\frac{\Delta \quad {{n_{0}(z)} \cdot {A_{c}(\alpha)}}}{2} \cdot \sin}\quad ( {{\varphi_{0}(z)} - {\alpha \quad z}} )} + {\frac{\Delta \quad {{n_{0}(z)} \cdot {A_{s}(\alpha)}}}{2}\quad \cdot \quad {\cos( {{\varphi_{0}(z)} - {az}}\quad )}}} ){{\alpha}.}}}} & (23)\end{matrix}$

[0068] In Equation (21), the first two terms on the right side representthe ideal grating, while Δn_(b)(z) and Δn_(r)(z) contain informationabout the defects (through the Fourier coefficients A_(c)(α) andA_(s)(α)) and behave as parasitic gratings, respectively at the blue andred sides of the spectrum with respect to the ideal reflection of thegrating.

[0069] Examples of such parasitic reflections are represented in FIG. 2(PRIOR ART) in the case of a linearly chirped grating. In FIG. 2, theposition z₁, corresponds to the grating front while z₂ corresponds tothe grating back and accordingly the grating provides a negativedispersion. As illustrated, the light of wavelength λ₁ coming from thegrating input is partly reflected at point z₁ by the blue side parasiticgrating, while the main reflection occurs at z₂ near the back of thegrating. Such a parasitic reflection affects the performances of thegrating. The light of wavelength λ₂ is reflected at location z₁, and bythe red side parasitic grating at z₂. Since a large part of the lighthas already been reflected at z_(l), especially if the grating isstrongly reflective, the parasitic reflection at z₂ has less effect whencompared to what occurs at wavelength λ₁. Thus, the front parasiticreflection, in this case at the blue side, is worse than the backparasitic reflection at the red side. Oppositely, the parasiticreflection at the red side is worse than the parasitic reflection at theblue side when the grating provides a positive dispersion, that is, whenz₂ corresponds rather to the grating front and z₁ to the back.

[0070] The blue parasitic reflection can be nulled by a proper phasecorrection function δφ_(a)(z). It will be noted that δφ_(a)(z) is aworking parameter corresponding to a hypothetical correction to theideal phase function δ₀(z), and has no relation to the phase errorfunction δφ(z) which is based on actual measurements of the grating.With such a correction, Equation (19) becomes:

n(z)=n_(eff0)(z)+Δn₀(z)·(1+δ_(n)(z))·sin(φ₀(z)+δφ_(a)(z)).  (24)

[0071] Assuming that δφ_(a)(z)<<1, Equation (24) reduces to:

n(z)=n _(eff0)(z)+Δn ₀(z)sin(φ₀(z))+Δn ₀(z)δ_(n)(z)sin(φ₀(z))+Δn₀(z)δφ_(a)(z)cos(φ₀(z)).  (25)

[0072] As was the case for the defect function δ_(n)(z), given byequation (20), the phase correction function δφ_(a)(z) can be expressedas Fourier spectra, that is: $\begin{matrix}{{{\delta\varphi}_{a}(z)} = {\int_{0}^{\infty}{( {{{B_{c}(\alpha)}{\cos ( {\alpha \quad z} )}} + {{B_{s}(\alpha)}{\sin ( {\alpha \quad z} )}}} )\quad {{\alpha}.}}}} & (26)\end{matrix}$

[0073] Combining Equations (20), (25) and (26), one obtains:

n(z)=n _(eff0)(z)+Δn ₀(z)·sin(φ₀(z))+Δn _(b)(z)+Δn _(r)(z),  (27)

[0074] where $\begin{matrix}{{{\Delta \quad {n_{b}(z)}} = {\int_{0}^{\infty}{\begin{pmatrix}{ {{{\frac{\Delta \quad {{n_{0}(z)} \cdot {A_{c}(\alpha)}}}{2} \cdot \sin}\quad ( {{\varphi_{0}(z)} + {\alpha \quad z}} )} - {\frac{\Delta \quad {{n_{0}(z)} \cdot {A_{s}(\alpha)}}}{2}\quad \cdot \quad {\cos( {{\varphi_{0}(z)} + {az}}\quad )}}} ) +} \\{{{{\frac{\Delta \quad {{n_{0}(z)} \cdot {B_{c}(\alpha)}}}{2} \cdot \cos}\quad ( {{\varphi_{0}(z)} + {\alpha \quad z}} )} + {\frac{\Delta \quad {{n_{0}(z)} \cdot {B_{s}(\alpha)}}}{2}\quad \cdot \quad {\sin( {{\varphi_{0}(z)} + {az}}\quad )}}}\quad}\end{pmatrix}{\alpha}}}},} & (28) \\{{\Delta \quad {n_{r}(z)}} = {\int_{0}^{\infty}{\begin{pmatrix}{ {{{\frac{\Delta \quad {{n_{0}(z)} \cdot {A_{c}(\alpha)}}}{2} \cdot \sin}\quad ( {{\varphi_{0}(z)} - {\alpha \quad z}} )} + {\frac{\Delta \quad {{n_{0}(z)} \cdot {A_{s}(\alpha)}}}{2}\quad \cdot \quad {\cos( {{\varphi_{0}(z)} - {az}}\quad )}}} ) +} \\{{{{\frac{\Delta \quad {{n_{0}(z)} \cdot {B_{c}(\alpha)}}}{2} \cdot \cos}\quad ( {{\varphi_{0}(z)} - {\alpha \quad z}} )} - {\frac{\Delta \quad {{n_{0}(z)} \cdot {B_{s}(\alpha)}}}{2}\quad \cdot \quad {\sin( {{\varphi_{0}(z)} - {az}}\quad )}}}\quad}\end{pmatrix}{{\alpha}.}}}} & (29)\end{matrix}$

[0075] The blue parasitic reflection can therefore be eliminated byadding a phase correction function δφ_(a)(z) such that, for all valuesof α:

B _(c)(α)=A _(s)(α),  (30)

B _(s)(α)=−A _(c)(α).  (31)

[0076] As developed in the previous section (Equations (15) and (17)), aphase correction function δφ_(a)(z) is equivalent to an average indexcorrection function Δn_(eff) _(—) _(ca)(z) given by: $\begin{matrix}{{\Delta \quad {n_{eff\_ ca}(z)}} = {{{- {n_{{eff}\quad 0}(z)}} \cdot \frac{p_{0}(z)}{2\pi} \cdot \frac{}{z}}{( {\delta \quad {\varphi_{a}(z)}} ).}}} & (32)\end{matrix}$

[0077] Alternatively, Δn_(eff) _(—) _(ca)(z) can be expressed as afunction of a period correction function δp_(a)(z): $\begin{matrix}{{{\Delta \quad {n_{eff\_ ca}(z)}} = {{\frac{n_{{eff}\quad 0}(z)}{p_{0}(z)} \cdot \delta}\quad {p_{a}(z)}}},} & (33)\end{matrix}$

[0078] where δp_(a)(z) is defined as: $\begin{matrix}{{\delta \quad {p_{a}(z)}} = {\frac{p_{0}^{2}(z)}{2\pi}\frac{}{z}{( {\delta \quad {\varphi_{a}(z)}} ).}}} & (34)\end{matrix}$

[0079] Finally, combining Equations (26), (30), (31) and (32), Δn_(eff)_(—) _(ca)(z) can be expressed as a function of the Fourier coefficientsof δ_(n)(z), that is A_(c)(α) and A_(s)(α) which are defined by Equation(20), so that one obtains: $\begin{matrix}{{\Delta \quad {n_{eff\_ ca}(z)}} = {\frac{{n_{{eff}\quad 0}(z)} \cdot {p_{0}(z)}}{2\pi}{\int_{0}^{\infty}{( {{{A_{c}(\alpha)}{\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}{\sin ( {\alpha \quad z} )}}} )\alpha {{\alpha}.}}}}} & (35)\end{matrix}$

[0080] Equation (35) therefore gives an average index correctioncompensating for the blue parasitic reflection, depending on theapodization defects function δ_(n)(z), through the Fourier coefficientsA_(c)(α) and A_(s)(α).

[0081] In cases where the red parasitic reflection needs to becompensated for, it can be seen from equation (29) that the signs on theFourier coefficients given by equations (30) and (31) need to bereversed to properly null the term Δn_(r)(z). This simply has the effectof changing the sign of the apodization correction function Δn_(eff)_(—) _(ca)(z), which therefore becomes: $\begin{matrix}{{\Delta \quad {n_{eff\_ ca}(z)}} = {{- \frac{{n_{{eff}\quad 0}(z)} \cdot {p_{0}(z)}}{2\pi}}{\int_{0}^{\infty}{( {{{A_{c}(\alpha)}{\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}{\sin ( {\alpha \quad z} )}}} )\alpha {{\alpha}.}}}}} & (36)\end{matrix}$

[0082] Post-Correction of the Defects

[0083] Based on the principles above, a first embodiment of the presentinvention therefore provides a post-correction method for improvingoptical properties of a Bragg grating using only an average indexcorrection.

[0084] The first step a) of this method involves characterising thephase defects, the apodization defects or both in the spatial refractiveindex profile of the Bragg grating. Since this particular embodimentrefers to a correction post-writing, it is understood that the gratinghas already been photoinduced in an appropriate photosensitive medium,and can therefore be characterised spectrally. Optical properties of thegrating are obtained, using routine measuring techniques. For example,any spectrum of the reflection, transmission, group delay either inreflection or transmission may be obtained, from either extremity of thegrating or both. Preferably, at least one reflectivity spectrum and onegroup delay spectrum in reflection are measured. The spatial refractiveindex profile of the grating is then reconstructed, based on themeasured optical properties. This is preferably achieved through the useof a reconstruction algorithm such as the Layer Peeling Method (see R.Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scatteringalgorithm for the design of nonuniform fiber Bragg gratings,” IEEE J.Quantum Electron., vol. 35, pp. 1105-1115 (1999), and J. Skaar, L. Wang,and T. Erdogan, “On the synthesis of fiber Bragg gratings by layerpeeling,” IEEE J. Quantum Electron., vol. 37, pp. 165-173 (2001)). Othercharacterization techniques based on direct measurement of theapodisation and phase profiles can be used as well (see P. A. Krug, R.Stolte, and R. Ulrich, “Measurement of index modulation along an opticalfiber Bragg grating,” Opt. Lett., vol. 20, pp. 1767-1769 (1995); M.Aslund, J. Canning, and L. Poladian, “Novel characterization techniquewith 0.5 ppm spatial accuracy of fringe period in Bragg gratings,” Opt.Express, vol. 11, pp. 838-842 (2003).

[0085] Referring to FIGS. 1A and 1B, an example of such a gratingcharacterization using this technique is shown in the case of adispersion compensation grating. Apodisation (FIG. 1A) and period (FIG.1B) profiles obtained from a Layer Peeling reconstruction of therefractive index profile are illustrated in comparison with thetheoretical profiles (dotted lines—a small offset has been added to thecurve of FIG. 1B to better see the differences between the superposedprofiles). It can be seen that both profiles are very close to thetheoretical curves. Also shown in FIGS. 1A and 1B are the respectiveprofiles' reconstructions from the data taken from both grating ends.Very similar results are obtained which is an indication that thereconstructed profiles properly describe the grating. In such a case,the profile from either ends may be used to characterize the defects. Inthe alternative, a properly weighted combination of the profiles fromboth ends could be computed as the final reconstructed profile.

[0086] The fluctuations of the reconstructed apodization profile fromthe theoretical curve correspond to the apodization defects functionδ_(n)(z) and the differences in the theoretical and reconstructed periodprofiles give the period defects function δp(z). The error function onthe average index δn_(eff)(z) is not characterised with this techniqueand is assumed to be zero. This assumption comes from the virtualredundancy that exists between the phase and average index errors asexplained above.

[0087] Once the defects have been properly characterised, and thedefects functions δ_(n)(z) or δp(z) or both are obtained, a second stepb) of calculating an average index correction Δn_(eff) _(—) _(tc)(z) tothe spatial refractive index profile of the grating is performed. Theaverage index correction depends on at least one of the two kinds ofdefects δ_(n)(z) and δp(z), as obtained in step a), through appropriatecorrection functions Δn_(eff) _(—) _(cp)(z) and Δn_(eff) _(—) _(ca)(z).In cases where both types of defects are considered, the average indexcorrection Δn_(eff) _(—) _(tc)(z) is a sum of Δn_(eff) _(—) _(cp)(z) andΔn_(eff) _(—) _(ca)(z). An offset term Δn_(eff) _(—) _(offset) is addedwhen necessary to the average index correction Δn_(eff) _(—) _(tc)(z) torender it positive for all values of z.

[0088] Preferably, based on the principles explained above, when neededthe period correction function Δn_(eff) _(—) _(cp)(z) is given byequation (18), repeated here for convenience: $\begin{matrix}{{{\Delta \quad {n_{eff\_ cp}(z)}} = {{{- \frac{n_{{eff}\quad 0}(z)}{p_{0}(z)}} \cdot \delta}\quad {p(z)}}},} & (37)\end{matrix}$

[0089] where z is the propagation axis of the grating, n_(eff0)(z) isthe target average effective index and p₀(z) is the target local period.

[0090] Again, as previously explained, when the apodization correctionfunction Δn_(eff) _(—) _(ca)(z) is needed, a frequency domain analysisof the apodization defects function is preferably performed. Anapodization correction function Δn_(eff) _(—) _(ca)(z) is calculated tocompensate for a front parasitic reflection in the frequency domain,which may be on the blue or red side of the ideal reflection from thegrating depending on the sign of the grating dispersion. The result ofthis analysis is an apodization correction function taking the form ofequation (35) or (36): $\begin{matrix}{{{\Delta \quad {n_{eff\_ ca}(z)}} = {{\pm \frac{{n_{{eff}\quad 0}(z)} \cdot {p_{0}(z)}}{2\pi}}{\int_{0}^{\infty}{( {{{A_{c}(\alpha)}{\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}{\sin ( {\alpha \quad z} )}}} )\alpha {\alpha}}}}},} & (38)\end{matrix}$

[0091] where z is the propagation axis of the grating, n_(eff0)(z) isthe target average effective index, p₀(z) is the target local period andA_(s)(α) and A_(c)(α) are Fourier coefficients of said frequency domainanalysis. The positive sign applies in the case of a negative gratingdispersion, and the negative sign for a positive grating dispersion.

[0092] The last step of the method according to the present embodimentconsist of c) applying the average index correction to the grating. Theaverage index correction is photoinduced in the photosensitive mediumover the Bragg grating, preferably by exposing the grating to aspatially variable dose of UV light along its propagation axis.

[0093] Referring to FIG. 6, an example of an optical system 10 toperform this operation is shown. The optical system 10 includes a lightsource 12 generating actinic radiation, preferably in the form of a UVlight beam 14. The light beam impinges transversally on the Bragggrating 16, photoinduced in an optical fiber 18, and is scanned alongits propagation axis. In the illustrated embodiment, this is achieved byappropriate re-directing and shaping optical components mounted on atranslation stage 22, but of course any appropriate scanning assemblycould be used. To create the proper UV profile as dictated by thecalculated average index correction, the intensity of the UV light beam14 is varied during the scanning preferably using a variable opticalattenuator 20. Any alternative methods to vary the strength of thephotoinduced profile may also be used, such as varying the scanningspeed of the beam 14, moving the fiber 18 instead of, or in combinationwith, the scanning of the light beam, etc.

[0094] It is also understood that the optical system of FIG. 6 isillustrated by way of example and that any other set-up allowing tophotoinduce a variable average index profile could alternatively beused. For example, the average index correction could be applied usingthe same optical system used to make the Bragg grating in the firstplace. Preferably, such a system includes a phase mask in the path ofthe light beam, diffracting the beam to generate the interferencefringes of the grating (see for example U.S. Pat. No. 5,367,588 (HILL etal)). The application of the correction however does not require anyphase mask, as no fringes have to be written in the photosensitivemedium but rather only an average index change. Instead of removing thephase mask, it can remain in place for convenience but the fringes itproduces can be washed out by vibrating the phase mask (see M. J. Coleet al, “Moving fibre/phase mask-scanning beam technique for enhancedflexibility in producing fibre gratings with uniform phase mask,”Electron. Lett., pp.1488-1490 (1995)), or by using a curvature in thewavefront of the UV beam (as for example explained in U.S. Pat. No.6,501,883 (PAINCHAUD et al)).

[0095] Pre-Correction of the Defects

[0096] In accordance with another aspect of the present invention, thepost-correction described in the previous section can be replaced by apre-correction when the grating defects are found to be systematic fromgrating to grating. In this case, one or many test gratings are made andserve to characterise the defects and allow calculating a correction.Then, the fabrication procedure is adapted according to the calculatedcorrection.

[0097] In accordance with a preferred embodiment for such apre-correction, there is provided a method for making an improved Bragggrating using an optical system generating systematic defects, whichincludes the following steps of:

[0098] a) making at least one test Bragg grating using this opticalsystem. The system is set up to produce a target spatial refractiveindex profile such as the ideal profile given by equation (1).

[0099] b) Characterising period defects and apodization defects of thetest Bragg grating, to respectively obtain a period defects functionδp(z) and an apodization defects function δ_(n)(z). The reconstructiontechnique explained above may be used, but the present invention is notlimited to such a technique. It is understood that although data from asingle grating may be sufficient to properly characterise the systematicdefects generated by the optical system it can also be advantageous touse several gratings and compare their results to obtain thisinformation.

[0100] c) Calculating an average index correction to the target spatialrefractive index profile, as a function of the period and apodizationdefects functions. Preferably, the calculations developed aboveresulting in equations (37) and (38) are used.

[0101] d) Calculating a corrected spatial refractive index profile usingsaid average index correction; and

[0102] e) Making the improved Bragg grating using the optical system setup to produce the corrected spatial refractive index profile.

[0103] The fabrication procedure can include the average indexcorrection function given by Equation (9) directly as the grating iswritten, avoiding the need of an additional post-correction step.Alternatively, for an easier fabrication, the correction can also beapplied on the period profile of the grating to be written using theequivalence between average index and period expressed by Equation (17).Then, the target period profile p₀(z) can be replaced by a correctedperiod profile p_(c)(z) given by:

p _(c)(z)=p₀(z)+δp _(tc)(z),  (39)

[0104] where $\begin{matrix}{{\delta \quad {p_{tc}(z)}} = {{\frac{p_{0}(z)}{n_{{eff}\quad 0}(z)} \cdot \Delta}\quad {{n_{eff\_ tc}(z)}.}}} & (40)\end{matrix}$

[0105] Referring to FIG. 7, there is shown an exemplary optical systemto write the test gratings and improved gratings according to apreferred embodiment of the invention. An example of an appropriatemethod to write complex period profiles using such a system is disclosedin U.S. Pat. No. 6,501,883 (PAINCHAUD et al), which is incorporatedherein by reference.

[0106] The optical system 110 first includes a phase mask 116 providedproximate the photosensitive medium 114 along the propagation axis 112.A light source 118, preferably a UV laser source, is also provided andgenerates a light beam 120 which is directed to project through aportion of the phase mask 116. This in turn generates a light beam witha modulated intensity profile which impinges on the photosensitivemedium 114 to locally record therein a portion of the optical gratinghaving a characteristic period.

[0107] Means for moving the light beam 120 along the propagation axis112 of the photosensitive medium 114, to successively record portions ofthe optical grating therealong, are further provided. In the illustratedembodiment, these moving means include a 45° mirror 122 disposed toredirect the light beam 120 from the light source 118 towards the phasemask 116, this mirror being mounted on a translation stage 124.

[0108] Similarly, means for moving the phase mask 116 in a directionparallel to the moving of the light beam 120 and concurrently theretoare provided. Preferably, this is embodied by a second translation stage126 on which the phase mask 116 is mounted. The relative movements ofthe phase mask 116 and the light beam 120 are adjusted to locally tunethe characteristic period of each portion of the optical grating.

[0109] Appropriate optical components forming an optical assembly 128may further be provided to give the light beam 120 a wavefront curvaturealong the direction of the waveguiding axis. The wavefront radius ofcurvature, in the plane of the phase mask, is selected to generallyoptimize the efficiency of the recording of the optical grating.

[0110] Experimental Results Obtained Using Preferred Embodiments of theInvention

[0111] Referring to FIGS. 3A (PRIOR ART) and FIG. 3B, there is shown theGDR as a function of wavelength respectively before and after theapplication of a post-correction to a chirped fiber Bragg grating,according to a preferred embodiment of the invention.

[0112] The application of the post-correction was done by superimposinga UV-induced spatially-dependent average index correction using ascanning technique. Varying the power of the UV beam during the scanningprocess allowed inducing a non-uniform index change. This process can becalibrated such that, for a given scan speed, a given UV power causes aknown index change.

[0113] As can be seen from a comparison of FIGS. 3A and 3B, theapplication of the correction visibly improved the optical properties ofthe grating. The amplitude of the GDR was reduced from 33 ps to 14 pspeak-to-peak which corresponds to more than a factor of 2 inimprovement.

[0114] A pre-correction was also successfully tested on a 4-channeldispersion compensation grating, having a dispersion of about −1500ps/nm. Three non-corrected gratings were written from which systematicGDR was observed. The measured properties of one of them are shown inFIG. 4 (PRIOR ART). Then a correction period profile was calculated fromthese gratings and applied to another grating from which thecharacterization, shown in FIG. 5 was obtained. Significant improvementsto the optical characteristics have been obtained, noteworthy on the GDR(18 ps peak-to-peak compared to 49 ps) and the reflectivity flatness(0.29 dB compared to 0.72 dB).

[0115] Of course, numerous modifications could be applied to theembodiments described above without departing from the scope of thepresent invention as defined in the appended claims.

1. A method for improving optical properties of a Bragg grating having aspatial refractive index profile along a propagation axis, comprisingthe steps of: a) Characterising defects of the spatial refractive indexprofile of said Bragg grating, said characterising comprising thesubsteps of: i. Measuring optical properties of said grating; ii.Reconstructing the spatial refractive index profile of said gratingbased on said measured optical properties; and iii. Comparing thereconstructed spatial refractive index profile with a target spatialrefractive index profile; b) Calculating an average index correction tothe spatial refractive index profile as a function of the defectscharacterised in step a); and c) Applying said average index correctionto the Bragg grating.
 2. The method according to claim 1, wherein stepa) comprises obtaining a period defects function δp(z).
 3. The methodaccording to claim 1, wherein step a) comprises obtaining an apodizationdefects function δ_(n)(z).
 4. The method according to claim 1, whereinstep a) comprises obtaining a period defects function δp(z) and anapodization defects function δ_(n)(z).
 5. The method according to claim1, wherein substep a) i. comprises measuring at least one reflectivityspectrum of said Bragg grating.
 6. The method according to claim 1,wherein substep a) i. comprises measuring at least one transmissionspectrum of said Bragg grating.
 7. The method according to claim 1,wherein substep a) i. comprises measuring at least one group delayspectrum in reflection of said Bragg grating.
 8. The method according toclaim 1, wherein substep a) i. comprises measuring at least one groupdelay spectrum in transmission of said Bragg grating.
 9. The methodaccording to claim 1, wherein substep a) i. comprises measuring opticalproperties of said grating from two opposite ends thereof.
 10. Themethod according to claim 1, wherein the reconstructing of substep a)ii. comprises using a reconstruction algorithm.
 11. The method accordingto claim 10, wherein said reconstruction algorithm is based on a LayerPeeling method.
 12. The method according to claim 2, wherein thecalculating of step b) comprises calculating a period correctionfunction Δn_(eff) _(—) _(cp)(z) given by:${\Delta \quad {n_{eff\_ cp}(z)}} = {{{- \frac{n_{{eff}\quad 0}(z)}{p_{0}(z)}} \cdot \delta}\quad {p(z)}}$

where z is the propagation axis of the grating, n_(eff0)(z) is thetarget average effective index and p₀(z) is the target local period. 13.The method according to claim 3, wherein the calculating of step b)comprises calculating an apodization correction function Δn_(eff) _(—)_(ca)(z) based on a frequency domain analysis of the apodization defectsfunction.
 14. The method according to claim 13, wherein said apodizationcorrection function compensates for a front parasitic reflection in saidfrequency domain.
 15. The method according to claim 14, wherein saidfront parasitic reflection is on a blue side of an ideal reflectionspectrum of said grating, and said apodization correction function isgiven by:${\Delta \quad {n_{eff\_ ca}(z)}} = {\frac{{n_{{eff}\quad 0}(z)} \cdot {p_{0}(z)}}{2\pi}{\int_{0}^{\infty}{( {{{A_{c}(\alpha)}{\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}{\sin ( {\alpha \quad z} )}}} )\alpha {\alpha}}}}$

where z is the propagation axis of the grating, n_(eff0)(z) is thetarget average effective index, p₀(z) is the target local period andA_(s)(α) and A_(c)(α) are Fourier coefficients of said frequency domainanalysis.
 16. The method according to claim 14, wherein said frontparasitic reflection is on a red side of an ideal reflection spectrum ofsaid grating, and said apodization correction function is given by:${\Delta \quad {n_{eff\_ ca}(z)}} = {{- \frac{{n_{eff0}(z)} \cdot {p_{0}(z)}}{2\quad \pi}}{\int_{0}^{\infty}{( {{{A_{c}(\alpha)}\quad {\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}\quad {\sin ( {\alpha \quad z} )}}} )\alpha \quad {\alpha}}}}$

where z is the propagation axis of the grating, n_(eff0)(z) is thetarget average effective index, p₀(z) is the target local period andA_(s)(α) and A_(c)(α) are Fourier coefficients of said frequency domainanalysis.
 17. The method according to claim 4, wherein the calculatingof step b) comprises the sub-steps of: i. calculating a periodcorrection function Δn_(eff) _(—) _(cp)(z) given by:${\Delta \quad {n_{eff\_ cp}(z)}} = {{{- \frac{n_{eff0}(z)}{p_{0}(z)}} \cdot \delta}\quad {p(z)}}$

 where z is the propagation axis of the grating, n_(eff0)(z) is thetarget average effective index and p₀(z) is the target local period; ii.calculating an apodization correction function Δn_(eff) _(—) _(ca)(z)based on a frequency domain analysis of the apodization defectsfunction, said apodization correction function compensating for a frontparasitic reflection in said frequency domain appearing on a blue sideof an ideal reflection from said grating, said apodization correctionfunction being given by:${\Delta \quad {n_{eff\_ ca}(z)}} = {\frac{{n_{eff0}(z)} \cdot {p_{0}(z)}}{2\quad \pi}\quad {\int_{0}^{\infty}{( {{{A_{c}(\alpha)}\quad {\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}\quad {\sin ( {\alpha \quad z} )}}} )\alpha \quad {\alpha}}}}$

 where A_(s)(α) and A_(c)(α) are Fourier coefficients of said frequencydomain analysis; and iii. calculating the average index correctionfunction as a sum of the period correction function, the apodizationcorrection function and a uniform index change offset selected to rendersaid average index correction strictly positive.
 18. The methodaccording to claim 4, wherein the calculating of step b) comprises thesub-steps of: i. calculating a period correction function Δn_(eff) _(—)_(cp)(z) given by:${\Delta \quad {n_{eff\_ cp}(z)}} = {{{- \frac{n_{eff0}(z)}{p_{0}(z)}} \cdot \delta}\quad {p(z)}}$

 where z is the propagation axis of the grating, n_(eff0)(z) is thetarget average effective index and p₀(z) is the target local period; ii.calculating an apodization correction function Δn_(eff) _(—) _(ca)(z)based on a frequency domain analysis of the apodization defectsfunction, said apodization correction function compensating for a frontparasitic reflection in said frequency domain appearing on a red side ofan ideal reflection from said grating, said apodization correctionfunction being given by:${\Delta \quad {n_{eff\_ ca}(z)}} = {{- \frac{{n_{eff0}(z)} \cdot {p_{0}(z)}}{2\quad \pi}}{\int_{0}^{\infty}{( {{{A_{c}(\alpha)}\quad {\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}\quad {\sin ( {\alpha \quad z} )}}} )\alpha \quad {\alpha}}}}$

 where A_(s)(α) and A_(c)(α) are Fourier coefficients of said frequencydomain analysis; and iii. calculating the average index correctionfunction as a sum of the period correction function, the apodizationcorrection function and a uniform index change offset selected to rendersaid average index correction strictly positive.
 19. The methodaccording to claim 1, wherein step c) comprises photoinducing saidaverage index correction in said photosensitive medium over said Bragggrating.
 20. The method according to claim 19, wherein step c) comprisesexposing the Bragg grating to a spatially variable dose of UV lightalong the propagation axis of said grating.
 21. The method according toclaim 20, wherein step c) comprises the sub-steps of: i. impinging a UVlight beam on said Bragg grating; ii. scanning said UV light beam alongthe propagation axis of the grating; and iii. varying the intensity ofthe UV light beam during said scanning to produce said average indexcorrection.
 22. A method for making an improved Bragg grating using anoptical system generating systematic defects, comprising the steps of:f) making at least one test Bragg grating using said optical system setup to produce a target spatial refractive index profile; g)Characterising period defects and apodization defects of said test Bragggrating to respectively obtain a period defects function δp(z) and anapodization defects function δ_(n)(z); h) Calculating an average indexcorrection to the target spatial refractive index profile as a functionof the period and apodization defects functions; i) Calculating acorrected spatial refractive index profile using said average indexcorrection; and j) Making said improved Bragg grating using said opticalsystem set up to produce the corrected spatial refractive index profile.23. The method according to claim 22, wherein the characterising of stepb) comprising the substeps of: i. Measuring optical properties of saidtest Bragg grating; ii. Reconstructing an apodization profile and aperiod profile of said test Bragg grating based on said measured opticalproperties; and iii. Comparing the reconstructed period and apodizationprofiles with target profiles thereof.
 24. The method according to claim23, wherein the reconstructing of substep b) ii. comprises using areconstruction algorithm.
 25. The method according to claim 24, whereinsaid reconstruction algorithm is based on a Layer Peeling method. 26.The method according to claim 22, wherein the average index correctioncalculated in step c) is given by: Δn _(eff) _(—) _(tc)(z)=Δn _(eff)_(—) _(cp)(z)+Δn _(eff) _(—) _(ca)(z) where Δn_(eff) _(—) _(cp)(z) is aperiod correction function and Δn_(eff) _(—) _(ca)(z) is an apodizationcorrection function.
 27. The method according to claim 26, wherein theperiod correction function is given by:${\Delta \quad {n_{eff\_ cp}(z)}} = {{{- \frac{n_{eff0}(z)}{p_{0}(z)}} \cdot \delta}\quad {p(z)}}$

where z is the propagation axis of the grating, n_(eff0)(z) is thetarget average effective index and p₀(z) is the target local period. 28.The method according to claim 26, wherein the apodization correctionfunction is based on a frequency domain analysis of the apodizationdefects function.
 29. The method according to claim 28, wherein saidapodization correction function compensates for a front parasiticreflection in said frequency domain.
 30. The method according to claim29, wherein said front parasitic reflection is on a blue side of anideal reflection from said test grating, and said apodization correctionfunction is given by:${\Delta \quad {n_{eff\_ ca}(z)}} = {\frac{{n_{eff0}(z)} \cdot {p_{0}(z)}}{2\quad \pi}\quad {\int_{0}^{\infty}{( {{{A_{c}(\alpha)}\quad {\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}\quad {\sin ( {\alpha \quad z} )}}} )\alpha \quad {\alpha}}}}$

where z is the propagation axis of the grating, n_(eff0)(z) is thetarget average effective index, p₀(z) is the target local period andA_(s)(α) and A_(c)(α) are Fourier coefficients of said frequency domainanalysis.
 31. The method according to claim 29, wherein said frontparasitic reflection is on a red side of an ideal reflection from saidtest grating, and said apodization correction function is given by:${\Delta \quad {n_{eff\_ ca}(z)}} = {{- \frac{{n_{eff0}(z)} \cdot {p_{0}(z)}}{2\quad \pi}}{\int_{0}^{\infty}{( {{{A_{c}(\alpha)}\quad {\cos ( {\alpha \quad z} )}} + {{A_{s}(\alpha)}\quad {\sin ( {\alpha \quad z} )}}} )\alpha \quad {\alpha}}}}$

where z is the propagation axis of the grating, n_(eff0)(z) is thetarget average effective index, p₀(z) is the target local period andA_(s)(α) and A_(c)(α) are Fourier coefficients of said frequency domainanalysis.
 32. The method according to claim 22, wherein the correctedaverage refractive index profile calculated in step d) is obtained byadding the average index correction to the target spatial refractiveindex profile.
 33. The method according to claim 22, wherein thecorrected average refractive index profile calculated in step d) isobtained by replacing the period profile of the target spatialrefractive index profile by a corrected period profile given by:${p_{c}(z)} = {{p_{0}(z)} + {{\frac{p_{0}(z)}{n_{eff0}(z)} \cdot \Delta}\quad {n_{eff\_ tc}(z)}}}$

where z is the propagation axis of the grating, p₀(z) is the targetperiod profile, n_(eff0)(z) is the target average refractive index andΔn_(eff) _(—) _(tc)(z) is the average index correction calculated instep c).